Simplify the following expression and state the condition under which the simplification is valid: $a = \dfrac{r^2 + 3r - 70}{r^2 + 19r + 90}$
Solution: First factor the expressions in the numerator and denominator. $ \dfrac{r^2 + 3r - 70}{r^2 + 19r + 90} = \dfrac{(r - 7)(r + 10)}{(r + 9)(r + 10)} $ Notice that the term $(r + 10)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(r + 10)$ gives: $a = \dfrac{r - 7}{r + 9}$ Since we divided by $(r + 10)$, $r \neq -10$. $a = \dfrac{r - 7}{r + 9}; \space r \neq -10$